From Marbles to Daxes

An Introduction to HBMs and their Application to Category Learning”

Jan Luca Schnatz

Hierarchical Beta-Binomial Model

Motivating Example

Repeatedly draw from bags of black and white marbles with unknown proportion of black marbles:

What color woud you predict for the next marble drawn from bag 8?
How did you arrive at that prediction?

Intuition

But think about that for a second. Mathematically, observing one black marble is incredibly weak evidence. If that was the only thing you saw, you shouldn’t be confident at all.
So why were you confident? Because you didn’t look at Bag 8 in isolation. You looked at the context.
Having seen that within each bag, colors tend to be homogenous (so either almost completely black or white) makes the single black marble highly informative
You noticed a pattern of Homogeneity: The previous bags were never “mixed.” They were always almost completely black or completely white.
This creates a strong Prior: “Bags tend to be uniform.”
Now, here is the most important part. It is not just about having “lots of past data.”
Imagine I took all those previous marbles and dumped them into one giant pile on the floor.
You would see a pile that is 50% black and 50% white.
If I drew one black marble from that pile, you would predict the next one is… well, probably 50/50.
So, the raw data didn’t change your mind. The structure did. The fact that the data came in bags is what allows us to learn the abstraction.

One black marble alone gives little information about the color future marbles
But seeing many mostly-black or mostly-white bags before makes that single black marble highly informative

\(\rightarrow\) High chance of next marbles also being black!

Intuition

So we have established that the grouping – the hierarchical structure – of the data is what matters for this inference
Because information at higher levels could be shared across the bags we developed a strong prior expectation of the marble color for the 8th bag after seeing the first marble, and thus we can make precise predictions from these priors
From the computational perspective of David Marr, we need to ask: What is the computational problem the mind is solving here?
Our goal is to reverse-engineer this intuition.
We want to construct a Bayesian model that captures this reasoning,
To solve this computational problem, we need to translate our “gut feeling” about the bags into a formal probabilistic model.
We will build this model from the bottom up, starting with the only thing the mind can actually observe directly

Hierarchy
- Information is shared across bags at higher levels
- Observations from previous bags shape strong priors
- These priors influence predictions about new bags

Goal

We want to build a Bayesian model that reverse-engineers the mind‘s reasoning about color distributions across bags.

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(\theta_i\): Probability of drawing a black marble in bag \(i\)
Different bags can have different probabilities \(\theta_i\)

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Level 3 – General knowledge about bags

\(\theta_i \sim \text{Beta}(\alpha, \beta)\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Level 3 – General knowledge about bags

Reparamerization as

Expected value \(\frac{\alpha}{\alpha + \beta}\) of \(\theta_i\)
Precision \(\alpha + \beta\) capturing concentration of probability mass around its mean (inverse of variance)

\(\theta_i \sim \text{Beta}(\alpha, \beta)\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Level 3 – General knowledge about bags

\(\theta_i \sim \text{Beta}(\alpha, \beta)\)

Level 4 – Hyperparameters

\(\frac{\alpha}{\alpha + \beta} \sim \text{Unif}(0, 1)\)

\(\alpha + \beta \sim \text{Exp}(1)\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Level 3 – General knowledge about bags

\(\theta_i \sim \text{Beta}(\alpha, \beta)\)

Level 4 – Hyperparameters

Prior of Beta Distribution
Uniform prior of \(\frac{\alpha}{\alpha + \beta}\) implies that every average probability of drawing a black marbles is equally likely prior to seeing any data
Exponential distribution of \(\alpha + \beta\) implies that smaller values are more likely prior to seeing any data

\(\frac{\alpha}{\alpha + \beta} \sim \text{Unif}(0, 1)\)

\(\alpha + \beta \sim \text{Exp}(1)\)

Formalisierung des Problems

We have \(i\) bags of marbles, where \(y_i\) is the number of black marbles observed and \(n_i\) is the total marbles drawn.

Level 1 – Data

\(d_i: \big\{y_i, n_i \big\}\)

Level 2 – Bag-specific distribution

\(y_i ~ \big| ~ n_i \sim \text{Binom}(\theta_i)\)

Level 3 – General knowledge about bags

\(\theta_i \sim \text{Beta}(\alpha, \beta)\)

Level 4 – Hyperparameters

\(\frac{\alpha}{\alpha + \beta} \sim \text{Unif}(0, 1)\)

\(\alpha + \beta \sim \text{Exp}(1)\)

Posterior Inference of HBM

[Top Equation: The Structure]

Now that we have formalized our intuition in a probabilistic model that captures that hierarchical structure of the data, the next question we should ask ourselfs: How do we estimate this model?
Well, we apply Bayes’ rule
The Posterior distribution is “simply” the product of the levels we just built: The Hyperprior (Level 4), the Conditional Prior (Level 3), and the Likelihood (Levels 1 & 2).

[Bottom Equation: The Mechanism]

The second equation is where the magic happens.
We want to predict \(\theta_i\) for a specific bag.
Since we don’t know the “true” \(\alpha\) and \(\beta\), we don’t just guess one value.
Instead, this integral averages over all possible values of \(\alpha\) and \(\beta\), weighted by how likely they are given the data (\(d_1 \dots d_n\)).
[Key Takeaway]

This is the mathematical definition of “Sharing Statistical Strength.”By integrating out \(\alpha\) and \(\beta\), the data from all previous bags flows down to influence the prior for the current bag

Applying Bayes Formula to HBM

\[ \begin{gathered} \overbrace{P(\theta, \alpha, \beta ~ | ~ y)}^{\text{Posterior}} \propto \underbrace{P(\alpha, \beta)}_{\text{Hyperprior}} \overbrace{P(\theta ~ | ~ \alpha, \beta)}^{\text{Conditional Prior}} \underbrace{P(y ~ | ~ \theta, \alpha, \beta)}_{\text{Likelihood}} \end{gathered} \]

Posterior inference regarding \(\theta_i\) by integrating out \(\alpha\) and \(\beta\)

\[ \begin{align*} P(\theta_i ~ | ~ d_1, \dots, d_n) = \iint P(\theta_i ~ | ~ \alpha, \beta, d_i) P(\alpha, \beta ~ | ~ d_1, \dots, d_n) \,d\alpha \,d \beta \end{align*} \]

Implementation

Numerical integration of equation approximated using Markov Chain Monte Carlo (MCMC) methods, e.g., Hamiltonian-Monte-Carlo (HMC) using STAN:

// Beta-Binomial Hierarchical Model in STAN
data {
  int<lower=0> N;           // Number of bags
  array[N] int<lower=0> n;  // Number of marbles drawn from each bag
  array[N] int<lower=0> y;  // Number of black marbles in each bag
}

parameters {
  real<lower=0,upper=1> mu;               // Hyperparameter: mean of the Beta distribution              
  real<lower=0> phi;                      // Hyperparameter: precision of the Beta distribution               
  array[N] real<lower=0, upper=1> theta;  // Bag-specific proportion of black marbles
}

transformed parameters {
  // Reparameterization of the Beta distribution
  real<lower=0> alpha = mu * phi;
  real<lower=0> beta  = (1 - mu) * phi;
}

model {
  mu  ~ uniform(0, 1);        // Hyperprior for µ     
  phi ~ exponential(1);       // Hyperprior for ϕ   
  theta ~ beta(alpha, beta);  // Conditional prior for θ
  y ~ binomial(n, theta);     // Likelihood
}

Applying the Model to the Marbles Example

These are the results of fitting the model we formalized using HMC with STAN to the marble data I showed you
We read the model hierarchy from left to right
- On the left you see a repetition of the marbles we obtained
- The middle panel shows the posterior distributions of \(\theta_i\) parameters for each bag (y-axis)
- The right panel shows the posterior distributions for the hyperparameters \(\mu\) and \(\phi\)
Figures show learning from the 7 previous bags of marbles incluences the inference about the new bag from which the single black marble was drawn
If we zoom in on the posterior distribution for \(\theta_8\) for the 8th bag, we see that the hierachical bayesian model captures our intuition \(\rightarrow\) high certainty that probability of drawing a black marble is near one for the 8th bag
How was the model able to make this inference? To find out we have to look that the level three inferences about the \(\alpha\) and \(\beta\) variables [Zoom in]
Posterior on \(\alpha + \beta\) is close to zero, which means that the learned the overhypothesis that the bags tend to be homogenous in color
Posterior on \(\alpha / (\alpha + \beta)\) is centered around 0.5, which means that roughly half of the marble in the population of bags will be black and half white.

Second figure:

To show what the model would predict, if it had seen different data, I additionally simulate a second marble scenario
What you can see in the left panel, two key aspects:
- On average there seems to be less than 50% black marbles across the bags (in the first example it was roughly 50%)
- The bags are now way less homogenous \(\rightarrow\) there tend to be both white and black marbles in each bag
The shift in the average probability of drawing a black marble is reflected in the \(\alpha / (\alpha + \beta)\), which is now shifted to the left of 0.5 \(\rightarrow\) model has inferred that there are fewer black marbles that white marbles across the entire population of bags
The posterior distribution of \(\alpha + \beta\) indicates that in comparison to the previous example, the parameters are relatively larger (greater than 1), which indicates that the bags of marbles are expected to contain a mix of black and white marbles (so they are heterogeneous)
This explains why now the model predicts in the posterior distribution for \(\theta_8\), that the probability of drawing a black marble in bag 8 is probably not close to 1, and additionally there is high uncertainty

Scenario I
Scenario II

Interim Summary

Key Takeaway

Marble example demonstrated that HBMs nicely align with our intuition how structured data can be used to form strong overhypotheses.

Why does this matter?

This abstract knowledge is what enables rapid learning from sparse data and one-shot generalization.

Application of HBMs to
Category Learning

Motivating Example

A mother points to an unfamiliar object lying on the counter and tells her child that this is a pen.

Question

By which features do children generalize the concept of a pen and recognize future instances of a pen as a pen?

In principle, the child could generalize the word to objects with the same material, same color, same texture, or simply objects lying on the counter
But empirically, children tend to generalize the new word to other objects that share the shape

Shape Bias

The expectation that members of a category tend to be similar in shape, which is learned by the age of 24 months (Smith et al., 2002).

Model Adaption

Overview of Changes
	Marble Example	Shapes Example
Structuring Variable	Bag	Object Category
Data	Marble	Object Exemplar
Features	Color	Shape, Color, Texture, Size, etc.
Feature Values	Binary	Categorical

Level 1: Binary Observations \(\rightarrow\) Categorical Observations
Level 2: Binomial Distribution \(\rightarrow\) Multinomial Distribution
Level 3: Beta Distribution Prior \(\rightarrow\) Dirichlet Prior
Level 4: Hyperprior similar to before

Copies of level 2-4. for each Feature Dimension (Color, Shape, Texture, Size)

Model Adaption

The model infers that categories are consistent in shape (low variance) but variable in color (high variance).
This learned structure creates a strong prior expectation that any new category will also be homogeneous in shape.
Consequently, the model enables rapid generalization of novel labels based on shape similarity, effectively ignoring differences in color.

Application to Noun Generalization Task

Glassen & Nitsch (2016) Griffiths et al. (2024) Kemp et al. (2007)

Table 1: Training Data

	1		2		3		4
Category	1	1	2	2	3	3	4	4
Shape	1	1	2	2	3	3	4	4
Texture	1	2	3	4	5	6	7	8
Color	1	2	3	4	5	6	7	8
Size	1	2	1	2	1	2	1	2

Two exemplar per category (columns)
Different feature dimensions (shape, texture, color size)
Pairs of objects belonging to the same category share the same shape!

Table 2: Testing Data

	'Dax'	Object 1	Object 2	Object 3
Category	5	?	?	?
Shape	5	5	6	6
Texture	9	10	9	10
Color	9	10	10	9
Size	1	1	1	1

After training, children (and the model) encounter a new object with a novel noun “dax”.

Task: Which of the three candidates with unkown label categories is most likely to be a dax?

Data based on Smith et al. (2002)

Results of Noun Generalization Task

19-month-olds who received the structured training choose the shape match
Untrained 19-month-olds choose randomly
The hierarchical Bayesian model shows the same preference pattern as trained children

Summary

Test

References

Glassen, T., & Nitsch, V. (2016). Hierarchical Bayesian models of cognitive development. Biological Cybernetics, 110, 217–227. https://doi.org/10.1007/s00422-016-0686-6

Griffiths, T. L., Chater, N., & Tenenbaum, J. (2024). Bayesian Models of Cognition: Reverse Engineering the Mind. MIT Press. https://mitpress.mit.edu/9780262049412/bayesian-models-of-cognition/

Kemp, C., Perfors, A., & Tenenbaum, J. B. (2007). Learning overhypotheses with hierarchical Bayesian models. Developmental Science, 10(3), 307–321. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-7687.2007.00585.x

Smith, L. B., Jones, S. S., Landau, B., Gershkoff-Stowe, L., & Samuelson, L. (2002). Object name learning provides on-the-job training for attention. Psychological Science, 13(1), 13–19. https://doi.org/10.1111/1467-9280.00403

From Marbles to Daxes

Hierarchical Beta-Binomial Model

Motivating Example

Intuition

Intuition

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Formalisierung des Problems

Posterior Inference of HBM

Implementation

Applying the Model to the Marbles Example

Interim Summary

Application of HBMs to Category Learning

Motivating Example

Model Adaption

Model Adaption

Application to Noun Generalization Task

Results of Noun Generalization Task

Summary

References

Application of HBMs to
Category Learning